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48
1. ALGEBRAIC THEMES
section are analogues of results of Section
1.6
, with the proofs also follow
ing a nearly identical course. In this discussion, the degree of a polynomial
plays the role that absolute value plays for integers.
Proposition 1.8.6.
Let
f
,
g
,
h
,
u
, and
v
denote polynomials in
KŒxŁ
.
(a)
If
uv
D
1
, then
u;v
2
K
.
(b)
If
f
j
g
and
g
j
f
, then there is a
k
2
K
such that
g
D
kf
.
(c)
Divisibility is transitive: If
f
j
g
and
g
j
h
, then
f
j
h
.
(d)
If
f
j
g
and
f
j
h
, then for all polynomials
s;t
,
f
j
.sg
C
th/
.
Proof.
For part (a), if
uv
D
1
, then both of
u;v
must be nonzero. If either
of
u
or
v
had positive degree, then
uv
would also have positive degree.
Hence both
u
and
v
must be elements of
K
.
For part (b), if
g
D
vf
and
f
D
ug
, then
g
D
uvg
, or
g.1
±
uv/
D
0
.
If
g
D
0
, then
k
D
0
meets the requirement. Otherwise,
1
±
uv
D
0
,
so both
u
and
v
are elements of
K
, by part (a), and
k
D
v
satisﬁes the
requirement.
The remaining parts are left to the reader.
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 Fall '08
 EVERAGE
 Algebra, Integers

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